2/7/2002, MATRICES AND GAUSS-JORDAN Math 21b, O. KnillHOMEWORK: 1.2: 6,11,18,20,30,38*, 1.3:4,14,26*,34,48,50, Due: Tue 2/12/2002MATRIX REFORMULATION. Consider thesystem of linear equations3x − y − z = 0−x + 2y − z = 0−x − y + 3z = 9The system can be written as A~x =~b, where A is a matrix(called coefficient matrix) and and ~x and~b are vectors.A =3 −1 −1−1 2 −1−1 −1 3, ~x =xyz,~b =009We also look at the augmented matrix (theseperators are there for clarity reasons only)B =3 −1 −1 | 0−1 2 −1 | 0−1 −1 3 | 9.MATRIX ”JARGON”. A rectangular array of numbers is called a matrix. If the matrix has m rows and ncolumns, it is called a m × n matrix. A matrix with one column only is called a column vector. The entriesof a matrix are denoted by aij, where i is the row and j is the column. In the case of the linear equation above,the matrix A is a square matrix and the augmented matrix B above is a 3 × 4 matrix.REWRITING A LINEAR EQUATION.A~x =− ~w1−− ~w2−. . .− ~wm−|~x|=~w1· ~x~w2· ~x. . .~wm· ~x=| | · · · |~v1~v2· · · ~vn| | · · · |x1x2· · ·xm= x1~v1+ x2~v2+ · · · + xm~vm=~b .In words: the entries biare the dot product of row vectors ~wiwith ~x. The vector~b is a sum of scaled columnvectors ~vjof A.GAUSS-JORDAN ELIMINATION. Gauss-Jordan Elimination is a process, where successive subtraction of mul-tiples of other rows or scaling brings the matrix into a so-called reduced row echelon form. The eliminationprocess consists of three possible steps which are called elementary row operations:• Swap two rows.• Divide a row by a scalar• Subtract a multiple of a row from an other row.The process transfers a given matrix A into a new matrix rref(A) in reduced row echelon form.REDUCED ECHELON FORM. A matrix is called in reduced row echelon form1) if a row has nonzero entries, then the first nonzero entry is 1. (leading one)2) if a column contains a leading 1, then the other column entries are 0.3) if a row has a leading 1, then every row above has leading 1’s to the left.RANK. The number of leading 1 in rref(A) is called the rank of A.SOLUTIONS OF LINEAR EQUATIONS. If Ax = b is a linear system of equations with m equations and nunknowns, then A is a m × n matrix. We have the following three possibilities:• Exactly one solution. There is a leading 1 in each row but not in the last row.• No solutions. There is a leading 1 in the last row.• Infinitely many solutions. There are rows without leading 1 and no leading 1 is in the last row.JIUZHANG SUANSHU. The technique of successively eliminating variables from sys-tems of linear equations is called Gauss elimination and appeared already in theChinese manuscript ”Jiuzhang Suanshu” (’Nine Chapters on the Mathematical art’).The manuscript appeared around 200 BC in the Han dynasty and was probably usedas a textbook. For more history of Chinese Mathematics, seehttp://aleph0.clarku.edu/˜djoyce/mathhist/china.html.EXAMPLES. The reduced echelon form of the augmented matrixB determines on how many solutions the linear system Ax = bhas.THE GOOD (exactly onesolution)THE BAD (no solution)THE UGLY (infinitely manysolutions)"0 1 2 | 21 −1 1 | 52 1 −1 | −2#"1 −1 1 | 50 1 2 | 22 1 −1 | −2#"1 −1 1 | 50 1 2 | 20 3 −3 | −12#"1 −1 1 | 50 1 2 | 20 1 −1 | −4#"1 0 3 | 70 1 2 | 20 0 −3 | −6#"1 0 3 | 70 1 2 | 20 0 1 | 2#"1 0 0 | 10 1 0 | −20 0 1 | 2#Rank(A) = 3, Rank(B) = 3.0 1 2 | 21 −1 1 | 51 0 3 | −21 −1 1 | 50 1 2 | 21 0 3 | −21 −1 1 | 50 1 2 | 20 1 2 | −71 −1 1 | 50 1 2 | 20 0 0 | −91 0 3 | 70 1 2 | 20 0 0 | −91 0 3 | 70 1 2 | 20 0 0 | 1Rank(A) = 2, Rank(B) = 3.0 1 2 | 21 −1 1 | 51 0 3 | 71 −1 1 | 50 1 2 | 21 0 3 | 71 −1 1 | 50 1 2 | 20 1 2 | 21 −1 1 | 50 1 2 | 20 0 0 | 01 0 3 | 70 1 2 | 20 0 0 | 0Rank(A) = 2, Rank(B) = 2.JORDAN. The German geodesist Wilhelm Jordan (1842-1899) applied the Gauss-Jordan method to findingsquared errors to work on surveying. (An other ”Jordan”, the French Mathematician Camille Jordan (1838-1922) worked on linear algebra topics also and is often mistakenly credited with the Gauss-Jordan process.)GAUSS. Gauss developed Gaussian elimination around 1800 and used it tosolve least squares problems in celestial mechanics and later in geodesic compu-tations. In 1809, Gauss published the book ”Theory of Motion of the HeavenlyBodies” in which he used the method for solving astronomical problems. Oneof Gauss successes was the prediction of an asteroid orbit.On the 1. January of 1801, the Italian astronomer Giuseppe Piazzi (1746-1826)discovered Ceres, the first and largest known asteroid in our solar system. Ceresis a rock of 914 km diameter. (The picture shows a photo in infrared). Gausswas able to predict the orbit of Ceres from a few observations. By param-eterizing the orbit with parameters and solving a linear system of equations(similar to one of the homework problems, where you will fit a cubic curvefrom 4 observations), he was able to derive the
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